Let $\bf A$ be a symmetric and positive definite matrix. Assume that we modify $\bf A$ so that some element $a_{ij}$ with $i \neq j$ becomes $a_{ij} + \varepsilon$, where $\varepsilon > 0$ (note that $a_{ji}$ also becomes $a_{ji} + \varepsilon$ so that the new matrix ${\bf \tilde{A}}$ is also symmetric). It is clear that the grand sum of ${\bf \tilde{A}}$ is greater than the grand sum of $\bf A$, i.e.,
$${\bf 1}^\top {\bf \tilde{A}} {\bf 1} > {\bf 1}^\top {\bf A} {\bf 1}.$$
Is it also true that the following inequality holds?
$${\bf 1}^\top \left({\bf \tilde{A}}\right)^{-1} {\bf 1} < {\bf 1}^\top {\bf A}^{-1} {\bf 1}.$$
Motivation
This question matters to me for showing that if there are signals $\theta_1, \dots, \theta_n$ following a joint normal distribution with mean $(\theta,\dots,\theta)$ and variance $A$, and $\theta$ follows an improper diffuse distribution, the posterior variance of $\theta$ conditional on $k<n$ signals for agent $i$ would increase if we take out one of the signals and introduce another one more correlated to hers.